Principal Local Ideals in Weighted Spaces of Entire Functions

Abstract

This paper deals with principal local ideals in a class of weighted spaces of entire functions of one variable. Let $\rho > 1$ and $q > 1$, and define ${E_I}[\rho ,q]$ (respectively, ${E_P}[\rho ,q]$) to be the space of all entire functions f of one variable which satisfy $|f(x + iy)| = O(\exp (A|x{|^\rho } + A|y{|^q}))$ for some (respectively, all) $A > 0$. It is shown that in each of the spaces ${E_I}[\rho ,q]$ and ${E_P}[\rho ,q]$, the local ideal generated by any one function coincides with the closed ideal generated by the function. This result yields consequences for convolution on these spaces. It is also proved that when $\rho \ne q$ a division theorem fails to hold for either space ${E_I}[\rho ,q]$ or ${E_P}[\rho ,q]$.